Vow, . lv.i%0. g. O.V PH YSICALLF SIMILAR SYSTEMS. 345

OiX PHYSICALLY SIM I LAR SYSTEMS;

ILLA'

STRATIOXS OFTHE I:SE OF DIMENSIONAL EQI:ATIOXS.

BV E. BI:CKI~CHAM.

I. The 3fosi Genera/ Form of Phys~ca/ Equations. —Let it be requiredto describe by an equation, a relation which subsists among a number ofphysical quantities. of n different kinds. If several quantitics of anyone kind are involved in the relation, let them be specifie by the value

of any one and the ratios of the others to this one. The equation will

then contain n symbols QI ~ . Q„,one for each kin«l of quant. ity, and

also, in general, a number of ratios r', r", ctc. , so that it may bc written

f('QI g. . - Q r' r" . . ) = o

Let us suppose, for the present only, that the ratios r do not vary

during the phenomenon described by the equation: for example, if the

equation describes a property of a material system and involves lengths,

the system shall remain geometrically similar to itself «luring any changes

of size which may occur. L'nder this condition equation (i) rc«juccs to

F(Q» Q ".Q.) = o

If none of the quantities involved in the relation has bccn ovcrl«xikcd,

the equation wi11 give a complete description of the relation subsisting

among the quantities represente«l in it, and will bc a complete equation.

The coefficients of a complete equation are «limcnsionless numbers, i. e.,

if the quantities Q are measured by an absolute system of units, the coef-

ficients of the equation do not depend on the sizes of the fundamental

units but only on the fixed interrelations of the units which characterize

the system and differentiate it from any other absolute system.

To illustrate what is meant by a "complete "equation, wc may consider

the familiar equation

—= constant,L=

in which p is the pressure, e the specific volume, and 8 the absolute

temperature of a mass of gas. The constant is not dimensionlcss but

depends, even for a given gas, on thc units adopted for measuring p, e,

and 8; the equation is not complete. Further investigation shov s that

346 F. BUCKINGHA M. tSECONDSERIESe

the equation may be written

in which the symbol R stands for a quantity characteristic of each gasand diA'ering from one to another, but fixed for any giicn gas when theunits of p, v, and 8 are fixed. M, 'e thus recognize that R is a quantity thatcan be measured by a unit derived from those of p, v, and 0. If wc rlo

express the value of R in terms of a unit thus dcrii cd, X is a dimcnsionless

constant and does not depend on the sizes of the units of p, v, and 0 butonly on the fixed relation v hich the unit of R bears to thc»i. The equa-tion is now a " complete ' equation.

Every complete physical cr~uatio» ('2) lias thc»iorc «pccific form

Z3IQ&"Q2'~ Q„'"= o. {3)

Such expressions as log Q or sin Q do not occur in physical equations; for

no purely arithmetical operator, except a simple numerical multiplic. r,

can be applied to an operand which is»ot itself a dimcnsionlcss number,

because we can not assign any definit meaning to thc rc«ult of such an

operation. The reason whj such an cxprcssiri» «s ()'-' ca» appc;ir, is thatQ' may be regarded as a syml&ol for the result of operating on Q l&v Q.

For example, when we write 2 = P, l-" is a symbol fr)r the result of oper-

ating on a length l by itself. M, e are rlirccterl tu take thc length l as

operand and " operate on it with the length l "by constructing on it as a

base, a rectangle of altitude l; and the result of this operation, which fixean area A, is represented by l'-'. Q hcnever functions that do not have

the form of the terms in equation {3)appear to occur in physical equations

it is invariably found upon examination that thc arguments of these

functions are dimensionless numbers.

2. Ink'roduction of Dimensional Conditions. —9, c hai e now to niake use

of the familiar principle, which sce»&s to have been first stated by Fourier,

that all the terms of a physical equation must have the same dimensions,

or that every correct physical equation is dimensionally homogeneous.

Let equation (g) be divided through by any one term and it takes the

formZVQ'~Q ~ . Q '"+ t = o

in which the N's are dimensionless numbers. In virtue of the principle

of dimensional homogeneity the exponents ci, a2, c„ofeach term of

equation (y) must be such that that term has no dimensions or that a

dimensional equation

[Q&"Q2" . Q. "1 = [tjis satished.

Voi.. IV.No. 4. OiV PH YSICALLF SIMILA R SYSTEMS.

Let II represent a dimensionless product of the form

Il = Q~"Q2" ' Q.'".

so that equation (4) may be written more shortly

ZXH+r =o.Since II is dimcnsionlcss, H is dimensionless; and furthermore, anyproduct of the form H& IH. -" . H; ' is also dimensionless. Hence if H&,

H, H; represent all the separate independent dimensionless productsof thc form {'6) xvhich can be made up in accordance ~vith equation (5)from the quantities Q, equation (~) may be ~vritten in the form

XXII& IH2'~ II'+ r = o

and still satisfy the requirement of dimensional homogeneity.

ow there are, so far as this requirement is concerned, no restrictions

on the number of terms, thc values of the coefficients, or the values of

the exponents. Hence the merely represents some unknown function

of the independent arguments III, II; and equation (8) may more

sli11ply bc wflttcilIt (III, H. ~ ~ H ) = o

By reason of thc principle of dimensional homogeneity, every complete

physical equation of the form (2) is reducible to the form (9) in which

[H,] = [H.] = . = [H;] = [i] (ro)

and the number i, of separate independent arguments of P, is the maximum

number of independent dimensionless products of the form (6) xvhich can

bc made by combining the n quantities Q&, Q - . Q„in diferent ways.

9, c have next to find the value of i. Let k be the number of arbitrary

fundamental units needed as a basis for the absolute system [Q,],

[Q„]by which the Q's are measured. Then in principle and if we

disregard thc practical considerations connected xvith the preservation

of standards, ctc., there is always, among the e units [Q], at least one

set of k which may be used as fundamental units, the remaining (n —k)

being derived from them.

Now each equation of the form (5) with a particular set of exponents

a (corresponding to a particular dimensionless product II) is an equation

to which the dimensions of the units [Q] are subject. But since (a —k)

of the units are derivable from the other k and the units are otherwise

arbitrary, it is evident that each equation of the form (5) is in reality

equivalent to one of these equations of derivation. There are therefore

(rr —k) equations of the form (5) and the number of products II which

E. BUCKI VCHA hf. tSECONDSERIES

appear as independent variables in equation (9) is

i=n —k.

Furthermore, if [Q&], [Q&] . [Q~] are k of the rr units which might beused as fundamental, the i equations (g) may be written

in which the P'~ represent gk, & (&„, r. e. , t)&e qu«ntiti«. » that are,temporarily, regarded «s «)«rive«I.

To make use of any one «if cqu;it i«iris ( l I '1 f«ir flrl«)illg the spec}fi formof the corresponding II, we replace each of the I(&]'» «n«i the [P] )ix theknown dimensional equivalent for it in terms of wh«tevcr set of ][I funda-

mental units (such as mass, length, time. etc, ) we may happen t«i find

convenient. The resultihg equ«ti«in cont«ins the k independent funda-mental units, and since both nicni)xr»;irc «if z«r«i dir»cnsi«ins, the ex-

ponent of each unit must vani»h. M, « th«r«f«~n «&)it, iiri k iriti«pen«lent

linear equations which suffic to dctcrniir&c the k ex~i«tn«r&t» «r&«i »o t«i fix

the form of the II in question. M, c h«ve still. howcvcr, onc «r)iitrarychoice left which it is so»betimes convcr&ient to r»«kc usc «if. iincc theII's occur in equation (9) «s «rguments of an indcterminatc furi«. ti«in Pand are subject only to the condition of )xiog dimensi«iril«»», when wc

have found the specific form of any one of the II'», wc are at li)icrty t«i

replace this by any function of it; for this furr«. tion will «ls«i )x dir»en»i«&n-

less and will be indepcndcnt of the rcm«ining II's. This rcni«rk cnablcs

us to dispense with fractional exponents, when they happen t«i rc»ult

from equations of the form (I r), an«i so to sir»plify the writing down of

our results.

3. Illustration. —To make the meaning of thc foregoing developments

more evident we may treat an example. Lct us suppose that we have

to deal with a relation which involves one quantity of each of the following

e = 7 kinds:Name. Symbol. Dimensions.

z. Force. . . . . . . . . . . . . . . . . . . . . . . . . . F2. Density. . . . . . . . . . . . . . . . . . . . . . p

g. Length. . . . . . . . . . . . . . . . . . . . . . . . D

4. Linear speed . . . . . . . . . 55. Revolutions per unit time. . . . . . . . n

6. Viscosity. . . . . . . , . . . . -. . . -. . . . . .

y. Acceleration. . . . . . . . . . . . . . -. . . . - g

Vo(.. IV.i%0. 4. 0Ã PHYSICALLY SIMILAR SYSTEMS. 349

Three fundamental units are needed, i. e. , k = 3, but they need not be[nt, I, t] for we could also use [F, p, S] or [p, np, ] or several other com-binations. On the other hand, such combinations as [I, S, n] or [S, n, g]coul(l not bc used.

Uc know by scrtion (2) that any relation whatever which involves

all the above see en quantities and no others, nlust be expressible by an

c(luati(in which can t&c reduce(l to the form

(9, a)

because n —k = 7 —g = 4.Io fin(] a specifi form of this equation, we select 3 of the quanti(ies

as fun(jamcntal and proceed to use equations (I I).I.«t lis, to start with, set

F = g, , p = ()-, D = ()3

tl(ir&g a lxl~~ihl( ~«t of fun(l'llll(lltal unit~ ~utficicnt for dcrii ing theI 1 h( ri. I'h( n 5. n, p, ".«ct «s P&, P-. P3. P4 «n(l u c hai e, corresponding

I() ('( fu. lt if)lli' [ I I ].

(ii. (I)

fronl wllich toI ilk ill@ t ht.'

()f fF. p, D, 5j

det«R1llll« tile A s, p s and y s.tir. t (~f th«. :c equations and substituting the dimensions

ill lt Wc hai c

[tn" I:t "" X nt"I *s~ X I X lt ') = [t]

and sinrc m, l, and I are ind«pendent. this can be satisfied only if al, pl,an(l y& arc related as shown by the equations

&i+Pl = 0,

3A+'YK+ ior ]Pl

2ai+ I =0,will be more convenient to

dimensionless equally well if

remaining three equations of

pD'n' y'- pD'gII II = ---' II'0 F r 3 F s i F

Kc thcrcforc have IIi = F ~p~DS, which

write and satisfy the condition of being

we square it and write I&1 = pD'S", F.If we follow a similar method with the

the sct (i i, a) wc have

35o E. BUCKIXGHA M. tSECONDSERIES.

and equation (9, a} takes the form

( pD S2 pD'n p pD g10F ' F ' Fp' F (9, b)

Our conclusion is that any equation which is the correct and completeexpression of a physical relation subsisting among seven quantities ofthe kinds mentioned is reducible to the form (9, b).

If [F, p, D] were the only triad that could be used as fundamental units

for the seven kinds of quantity, equation (9, b) would be the only general

form of the equation; but in reality several other triads can be used, sothat other equations may be found which, while essentially equivalent to

(9, b), present a different appearance. If, for instance, we select thetriad [p, D, S], a process like that which led to equation (9, b) gives us

the equation& pD'P Dn pDS Dg ~

F ' S' p,' S'i (9 c)

to which we shall have occasion to refer later.4. The General Form to Which Any Physical Equation is Reducible. —

Equation (9), subject to equations (rt), gives the necessary form of any

relation which subsists among n quantities of different kinds: it is the

final form to which the dimensional conditions reduce equation (2).Now equation (2) describes a particular form of the more general relation

described by equation (r), in which several quantities of each of the n

kinds may be involved, —all but one of each kind being specified by their

ratios to that one. Dimensional reasoning can not furnish any informa-

tion regarding the influence of these dimensionless ratios on the phenome-

non which is characterized by the relation in question, nor can it tell

us how they are involved in the equation which describes the relation.

But we can not assume that they are without influence, and the possi-

bility of their entering into the relation must be indicated in the final

equation which corresponds to (r) as equation (9) does to (2). Since

equation (9) follows from equation (2), it is correct for any fixed values

of the r's, and it may therefore be generalized so as to be applicable to

any and all values of the ratios r by introducing the r's as independent

arguments of the unknown function P, which is then a function of all the

independent dimensionless combinations of powers of all the quantities

of all the n kinds which are involved in the relation to be described.

The general conclusion from the principle of dimensional homogeneity

may therefore be stated as follows: If a relation subsists among any

number of physical quantities of n different kinds, and if the symbols

Q&, Qn, Q„represent one quantity of each kind, while the remaining

Vog. . IV.No. 4. ON PHYSICALLY SIMILAR SYSTEMS.

quantities of each kind are specified by their ratios r', r", ~ ~, etc. , tothe particular quantity of that kind selected, then: any equation which

describes this relation completely is reducible to the form

|t(ili, II2, Il;, r', r" ) = o.

If k is the number of fundamental units required in an absolute systemfor measuring the n kinds of quantity, the number of the dimensionless

products II isi=n —k.

If [Q~], [Qsl [Qal are any k of the units for measuring the Q's, which

are independent and so might be used as fundamental units; and if theremaining units needed are denoted by [P&], [P,] [P;l, each of theII's may be determined from a dimensional equation

(t4)

after substituting in this equation the known dimensions of the [Q]'s and

of [P] in terms of any suitable set of k fundamental units.

5. Remarks of the Utilieatiort of the Foregoing Results. —Fquation (t3),representing a single relation connecting a number of variables, can, in

principle at least, be solved for any one of them and put into the form

in which IIi is any one of the II's; or into the form

r' = rp (II, Iis, II;, r", r'", .) (r6)

in which r' is any one of the r's.

Although the form of p is unknown so that neither of these equations

gives any definite general information, they may nevertheless be useful

in particular circumstances. Equation (rg), for example, tells us that

if II2, II~, ~ ~ ~, II;, r', r", etc. , are all kept constant, IIi is also constant,

regardless of the form of the unknown function p. And since IIi is a

product of known powers of Q&, Q2, . Q&, P|, we know how any one of

these (k + t) quantities varies with the others under the given conditions.

To illustrate; equation (9, b) may be put into the form

pD'S' l pD'rts pm pD" g 1

F i F ' Fp' F

and if pD'n'/F, p'/Fp and pD'g/F are kept constant we have

(tI)

F5' = const. XpDZ

(t 8)

352 E. 8UCKI1V GHA M. tSECONDSEXES.

If we wish to treat some one quantity X as the unknown and get anequation of the form (t5) with this quantity absent from the second

member, X must be a factor of only one of the II's. This means that in

selecting the variables which are to act as Q's and P's in equations (It)or I'rg), X must be one of the P's.

To illustrate the application of the above remarks, let us consider thescrew-propeller problem. Let I'" be the thrust exerted by a screw propeller

of a particular shape specified by a number of ratios of lengths r', r",etc. and of a size specihed by the diameter D. The thrust must be

supposed to depend on the number of revolutions per unit time n, the

speed of advance 5, and the density p and viscosity p, of the liquid. Itmay safely be assumed that the very slight compressibility of the liquid

has no sensible effect on the thrust, but unless the propeller is very

deeply immersed, there will be surface disturbances and we must expectthe thrust to be affected by the weight of the liquid, i. e., by the intensityof gravity g. It does not appear that any other circumstance, exceptthe depth of immersion which may be specified by its ratio to D and

represented by an extra r, can inHuence the thrust, and if we are right

in this assumption there must be an equation

f(F, p, D, S, n, p, g, r', r", ~ ) = o

corresponding to equation (t). This equation must be reducible to the

form (t3), and we have already given in equations (9, b) and (9, c), two

of the forms which it might have in the case of constant r's, ~. e. , for a

propeller of 6xed shape and immersion.

Now suppose that we wish to hand out how the thrust depends on the

density of the liquid, the diameter of the propeller, and the speed of

advance. Then in using equations (t t) we must use F as one of the P's

and take [p, D, S] as [Q&, Q2, Q3j. KVe then get the II's which appear in

equation (9, s) and the equation corresponding to (t3) is

& pD'5 Dn pD5 Df;p

By solving for pD'S /F, this may be put into the form

& Dn pD5 Df,S„1—,—,—,r', r (ao)

which tells us without any experimentation at all, that if we can keep

Dn/S, pDS/p, and Dg/S constant, the thrust of a propeller of any given

shape (r', r", etc. , constant) is proportional to the density of the liquid,

the square of the diameter, and the square of the speed of advance. The

Vor.. IV.No. 4. ON PH YSICA LL Y SIMILA R SYSTEhfS.

meaning of this result and the value of the infor mation willbe discussedlater: at present we may return to generalities.

6. Physically SimilarSystems Eq.—uation(r3) isaconvenientexpressionof the conclusions to be drawn directly from the principle of dimensional

homogeneity. It is useful in various ways, as will be illustrated later,but at present we may develop from it the notion of similar systems.

Let S be a physical system, and let a relation subsist among a number

of quantities Q which pertain to S. Let us imagine S to be transformed

into another system S' so that S' " corresponds" to S Bs regards theessential quantities. There is no point of the transformation at which

we can suppose that the quantities cease to be dependent on one another;hence we must suppose that some relation will subsist among the quan-

tities Q' in S' which correspond to the quantities Q in S. If this relation

in 5 is of the same form as the relation in S and is desrrib able by the

same equation, the two systems are " physically simil~ " as regards tlIis

relation. 9,'e have to enquire what sort of transformation would lead

to this result, ~. e., what are the conditions which determine that two

systems shall be similar as regards a given physical relation.The original relation subsisting in S is reducible to the form (i3), or

f(IIi, , II;, r) = o, (i3)

r representing all the independent ratios of quantities of the same kind

which enter into the relation. The changes of the Q's during the trans-

formation will, in general, result in a change of the numerical value of

each II or r. But these expressions remain dimensiohless, so that toeach of the arguments of P there corresponds, after the transformation,

an expression II' or r' of the same form in terms of the transformed quan-

tities Q', and these are all the independent dimensionless products of

powers that can be made up of the quantities Q'. Hence the equation

which describes the relation subsisting in S' among the quantities Q' is

reducible to the form

P'(ll', ~ ~ ll';, r') = o. (i3 )

The requirement that 5 and 5' shall be similar as regards this relation,

means that the operators P and P' must be identical, and this will occur

if the transformation leaves the numerical values of all the II's and r's

unchanged. For P and P' wi11 then be applied to identical operands:

and while two diHerent functions of the same variables may vanish

simultaneously for discrete sets of values of the variables, they can not

do so for a continuous infinity of sets, and yet equations (r3) and (13 )are satis6ed without restriction. It follows that f and P' can not be

E. BUCKI NGHA M. tSECOhDSERIES.

diAerent, or in other words; the systems S and S' are similar as regardsthis relation if corresponding II's and r's are equal in the two systems.The nature of the transformation which leaves a system similar to itselfmay therefore be specified as follows:

(a) Any k quantities of independent kinds may be changed in com-pletely arbitrary ratios, after which we must

(b) Change one quantity of each of the (n —k) = i remaining kindsin such a ratio as to keep the numerical value of its II unchanged; andfinally we must

(c) Change the remaining quantities of each of the n kinds in the sameratio as the one quantity of that kind already mentioned, therebykeeping the ratios r unchanged.

The last and simplest of these conditions means that the system mustremain similar to itself as regards each separate kind of quantity. If,for example, the sizes and shapes of some of its parts are essentiallyinvolved in the relation, the transformation must leave the system geo-metrically similar to itself as regards these parts, although other andunessential parts may change in any way. We have, in all, k arbitrarychoices of ratios of change and since each of these may be made in aninfinite number of ways there is a k-fold infinity of systems S which aresimilar to any given system S as regards any particular physical relation.If the above conditions are fulfilled for all possible physical quantitieswhich can pertain to a physical system, the transformed system will

be similar to the original one as regards any possible relation between

physical quantities and the two will be physically similar in all respects.When absolute units are used, the validity of a complete physical

equation is unaffected by changes in the fundamental units. Hence in

changing from a system S to a similar system S' it is immaterial to thevalidity of the equation in question whether we do or do not retain ouroriginal fundamental units. If we alter the sizes of the fundamentalunits [Q&] . . [Qz] in the same ratios as the kinds of quantity Q& Q&

which they measure, the numerical value of any quantity of one of thesekinds will be the same in both systems. And if we do not change therelations of the derived and fundamental units of our absolute system,every derived unit [P] will change in the same ratio as every quantity Pof that kind, so that the numerical value of every quantity in the systemS will be equal to the numerical value of the corresponding quantity in

the similar system S'~

This change, af units will occur if the concrete primary standards which

preserve the units partake of the transformation. To an observer whose

quantitative information was all obtained by measurements based on

Vot. . IV.No. 4. OV P1iYSICALLY SIS1ILAR SE STI'-.11S.

such standards, not only all physical relations, but the numerical valuesof individual quantities, would appear the same in two similar systems:he could not distinguish the two systems nor detect a transformation ofone into the other.

The foregoing theorem may, if we choose, be applied to an inwginarytransformation of the whole physical universe, but in this grandiosegeneral form it is of only metaphysical interest; for it is merely a state-ment about what would happen if we were to bring about certain changeswhich it is obviously quite beyond our powers to effect. Neverthelessin particular elementary instances the notion of physical similarity isuseful and it is convenient to have the conditions of physical siniil;~rityformulated in a general way. One of these is always that of similaritywith respect to each separate kind of quantity, such as length, speed,density, etc. , which may enter into the physical phenomenon with whichwe happen to be concerned.

Let us suppose that a relation subsists among certain physical quan-tities. Dimensional reasoning sukces to tell us that if the relation iscomplete, the equation which describes it is reducible to the form (r3);and if Q&

. Qq are the quantities which are used as independent in

finding the II's of equation (r3), the equation can always be solved inthe form ', = Q,.Q, Q,"~(n„n„n;,r)

the form of the operator q remaining to be found by other means. IVith-out going into any abstruse consideration of conceivable modifications ofthe universe to which the various quantities may be regarded as pertain-ing, it is obvious that so long as we can, experimentally, control enoughof these quantities to keep all the II's and r's constant, any function ofthese arguments must also remain constant, no matter what its form

may be. The practical application of the notion of similarity is basedon this remark. The conditions necessary for this simplification aregiven by setting each II and r equal to a constant: and when this is donethe nature of the possible simultaneous variations which fulfil the re-

quirements at once becomes evident.It usually happens that some of the quantities concerned in the relation

are obviously attached to or are properties of some body or some materialsystem of limited extent and can be changed in value only by changingto a different body or system. If this second system is similar to thefirst as regards each separate kind of quantity all the r's which pertainto the system are the same. If we can so arrange the circumstances in

which the system is placed that upon substituting one body or systemfor the other, II2, II3, ~ ~ - II; as well as any remaining r's, retain their values

unchanged, the equation

I'i = Q»'Q»» Q»» X const

is satisfied for both systems with the same value of the constant. Thephenomenon characterized by the relation then. occurs in a similar manner

for both systems, and we say that the bodies or systems are similar with

respect to this phenomenon. If the relation is a dynamical one, all

essential parts of the two systems must be geometrically similar and have

similar distributions of density, elasticity, etc. , so far as these propertiesacct their behavior. If, in. addition, the II's of the relation are kept thesame for one system as for the other, the systems are said to be "dy-namically similar, " though they might, of course, not be similar as regards

some other dynamical relation nor behave similarly in some different

sort of experiment.The notion of physical similarity does not appear to have been developed

and used to any extent except in this most obvious form of dynamical

similarity. But the more general conception of a similarity which

extends to other than. merely dynamical relations, evidently follows

directly from the dimensional reasoning, based on the principle of

homogeneity, which culminated in equations (t3) and (tg).7. Remarks. —In his article entitled "The Principle of Similitude, "

appearing in the April, I9I4, number of the PHYsIcAL REvIEAv, Mi.Richard C. Tolman announces the discovery of a new principle and

illustrates its value in reasoning about the forms of physical equations,

by treating several examples. The statement of the principle is couched

in such general terms that I have dif6culty in understanding just what

the postulate is, but it seems to me to be merely a particular case of the gen-

eral theorem given in the foregoing section. Mr. Tolman selects length,

speed, quantity of electricity, and electrostatic force as the four inde-

pendent kinds of quantity which suSce for his purposes, and after

subjecting them to four arbitrary conditions, he proceeds to find the

conditions to which several other kinds of quantity are subject in passing

from the actual universe to a miniature universe that is physically similar

to it. Now I do not know whether the developments set forth above

have ever been published in just this form, but it is certain. that they are

merely consequences of the principle of dimensional homogeneity, which

is far from being either new or unfamiliar. The unnecessary introduction

of new postulates into physics is of doubtful advantage, and it seems to

me decidedly better, from the physicist's standpoint, not to drag in either

electrons or relativity when we can get on just as well without them.

Accordingly, my object in publishing the foregoing sections, which are a

Vor. . IV.No. 4. ON PH YSICA LL Y SIMILAR SYSTE.11S. 357

fragment of a longer paper that I have had in hand for some time, is tocall attention to the fact that in the present instance no new postulateseems to be needed. My feeling that Mr. Tolman's "Principle ofSimilitude " is not really new may, of course, be mistaken. But for thepurposes to which he puts it, it is, at all events, superfluous, and this Ishall proceed to prove by treating some of the problems he has used asillustrations.

The relations that involve temperature may be passed over, becauseMr. Tolman's reasoning is based on the assumption that absolute temper-ature has the dimensions of energy, and this assumption is not permissible.If by

" absolute temperature " is meant temperature measured by what iscommonly called the thermodynamic scale, then the ratio of two temper-atures is, by definition, the ratio of two quantities of heat. In a similar

way, two intervals of time, measured b~ our ordinary time scale, havethe same ratio as two angles through which the earth has rotated aboutits axis during these intervals; or two forces have the same ratio as thelengths by which they can stretch a given spiral spring. Ke do not,however, conclude that time has the dimensions of angle and force thedimensions of length; nor can we say that temperature has the dimen-sions of energy. The units needed for measuring thermal quantitiescan not all be derived from mass, length, and time or from any other setof three fundamental units which suSce for niechanics, and a fourth unitis indispensable. In practice, this special thermal unit is nearly alwaystemperature; it is fixed by the arbitrary selection of the interval betweenthe freezing and boiling points of water "nd the arbitrary assignment of aparticular numerical value to this interval. 9,'e do not at present know

of any method by which this or any other interval of temperature canbe fixed by, i. e. , derived from, purely mechanical quantities e ithout some

further act of arbitrary choice than the selection of three mechanicalunits. We may therefore turn to Mr. Tolman's electromagnetic problems.

For the measurement of electric and magnetic quantities, one new

fundamental unit is needed, beyond the three of mechanics, so that there

must, in general, be four in all. In the electromagnetic system the new

unit is permeability [p] and the four are [m, l, t, p]. In the electrostaticsystem it is dielectric inductivity [e], and the four are [m, l, t, ~]. Other

sets are sometimes more convenient, for example [l, t, C, R], [C] being

current and [R] resistance: this system corresponds to the " international "units, a system in which the ampere and the ohm are, by definition, funda-

mental units. Various systems might be used in dimensional reasoning

without altering anything but a little algebra; the only important thing

is the number of fundamental units required as the basis of the system.

3S8 E. BUCKI.VGHA M.rSEcovnSERIES.

8. Illustrations of the Treatment of Electromagnetic Problems by the

ilfethod of Dimensions: (A) Energy Density of an Electromagnetic Field

XVe assume that the energy density u is completely determined by thefield strengths E and H, and by the permeability p and dielectric induc-

tivity ~ of the medium. If, or when, this assumption is valid we have

f(u, E, H, u, c) =o.

The dimensions of these quantities on the fm, l, t, u] system are

[E] = [m&ltt —'ts&], [u] = [ml —'t —']

[H] = [m'I-&t-'-u&], [e] = [l 't'u '].--KVe wish to get a relation that can be solved for u; hence u must not

be one of the Q's of equation (t4). Although in general, electrogmagnetic

units require four fundamental units, three are enough in this instance:for example we may take [E], [u], and [e] as fundamental units and from

them derive the remaining two units

With n = 5 and k = 3, n —k = i = 2 and there are only two of theI'I's. To determine them we have by equation (r4)

(a4)

To determine the exponents, we might substitute the dimensions given

in equations (22); but since we already have the dimensions of [H] and

[u] in terms of [E, e, u] by equations (2S), it is easier not to refer back to

the complicated [m, l, t, u] equations but use the [E, e, u] system at once,

Equations (24) then give us

(2S)

From the first of these we obtain the values

CXy = —I,

so that we may write

p1 1Pl

vot. . Iv.No. 4. ON PHYSICALLY SlMILAR SYSTEMS.

or more conveniently

p,H'-Hg =

From the second of equations (25) we have

whencey2 = Os

and the equation which corresponds to (9) or (I3) therefore has the form

r pII2 gg

(26)

Solving this for u/eE' and multiplying by ~E" we have-, finally,

For the sake of illustration we h«ve chosen to obtain this result bymeans of the general process described in the earlier sections: but theresult is obvious without the &id of «ny such el«bor«te m«chincry. Forsince by equation (23) [u] = [~E'], it is evident directly from thc principle

of dimensional homogeneity that if u is to be expressed «s « function of ~

and E it can only be in the form eE"- multiplied by a dimensionless

number.

By taking [H, s, u] as fundamentai, instead of [E, ~, u] we should have

got the obviously equivalent result

Assuming that the complete formula is

Iu = (eE'+ uH')—

8m.

we haveI +X

s i(n) = sr(~) =8~

If the medium is not isotropic, certain angles which fix the directions

of E and II with respect to the principal axes of e and p, must also appear

as arguments of the unknown functions y~ and q2.

(8) Relation between Mass and Radius of an Electron There is no.—object in limiting our considerations to a particular kind of disembodied

charge moving in free space, and we may as well make the treatment

more general.

36o E. BUCKINGHAM. tSECOVOSERIES.

Let T be the energy of a charge e of any fixed distribution and of asize specified by any one of its linear dimensions D, when it is moving atthe speed S through a medium of permeability p, and inductivity ~. Ifr', r", etc. , are ratios of lengths which specify the distribution of the charge,and if we assume that, for a fixed distribution, T does not depend on anyother quantities than those already named, we must have

f(T, e, D, S, p, e, r', r", ) = o. (28)

For measuring all these quantities, an absolute system requires fourfundamental units, the only new electrical quantity not mentioned inequations (22) being e, which has the dimensions

[e] = [m&l&p—

&].

Since we wish finally to solve for T, this must be one of the [P]'s ofequation (s4) and for the [Q]'s we must take four of the five units[e, D, S, p, e]. The five different combinations of these units, taken fourat a time are

D, S, e, ~; D, S, e, p, , D, S, e, p, D, e, e, p, S, e, e, p.

But since [pe) = [S '] as is seen from equations (22) S, e, and p can notbe used simultaneously as fundamental units, so that our choice islimited to the combinations

D, S, e, e, D, S, e, p, , D, e, e, p.

The three separate sets of units are then as follows:

x. (D, S, e, e], [p] = [S 'e '] [T] = [D 'e'e ']

2. D, S, e, p], [e] = [S'p']

and we can get a solution by using any one of these sets in equation (z4)for finding II~ and II2.

In any case, it is easily seen that S'pe will be one of the two II's. Theremaining II is to be found from one of the equations

[Il] = [D'See"e'D 'e'e '] = [t], —-[II] = (D See"p'D 'Pe'y] = [i), —

[II] = [D eee"p'D 'e'e '] = [t].

From these three equations we get successively

DeT DT DeTII = II= —, II=e' ' S'e'p' e'

Vol. . IV.No. 4. OX PHYSICALLY SIhfILAR SVSTEhfS. 361

Since the first and third solutions are identical we have only two differentforms of equation corresponding to (r3), and they are

li DeT II

or solving for T,

)t DT, Spa, r, r ~ i=o,

2

T = —— p|(S2pe, r', r" ),De

S'e'pT = —s,(S'pe, r', r", ).

(32)

(33)

(34)

It is interesting to consider the physical meaning of these results. EVc

did not restrict T to being merely energy duc to motion: it is the totalenergy and it must therefore reduce to To, the electrostatic energy of thecharge, when S = o. Equation (33) accordingly gives

2

T, =- —p, (o, r', r", )

and the agrees, as it should, with the known fact that the work done in

collecting a charge e, in a medium of inductivity ~, into a distribution oflinear size D is proportional to e2/De, the proportionality factor depending

on the shape of the distribution, i. e., on the ratios r', r", etc. Sinceequation (34) must give the same result, rp2 must contain the factor

t/S pe in order to keep T finite as S vanishes. If this factor is taken outthe two equations become identical ~

If we had originally let T represent only T„the part of the energydue to motion or the work required to start the charge going from rest,

nothing in the reasoning would have been changed arid thc resulting

equations would have had the same form as above. I f we define T,/5-" =m,

as the electrical mass of the moving charge, equation (34) gives us

and if We Set p, e = C-2, We have

(3S)

If we now let the charge be an electron of fixed quantity moving in free

space so that c, p and c are constant, the equation takes the simpler form

e, = f(S, r, —r, ) (36)

362 E. BUCKINGHAM. SBCOXOSBR1BS,

or in words: the electrical mass of an electron of fixed distribution is

inversely proportional to its linear dimensions; it also depends on thespeed and on the distribution, but the nature of this dependence is notdeterminable from dimensional reasoning.

(C) Radeahon from an Accelerated Electron —L.et a disembodied

charge e be moving with the velocity S through a medium of permeability

p, and inductivity». Let the distribution of the charge be specified bythe length ratios r', r", etc. , and its size by some linear dimension D.Let the charge have a resultant acceleration a which makes an angle 8

with S, 8 being dimensionless like the r's. Let R be the total time rateof loss of energy by radiation.

So far as we can tell a priori, R may depend on all of the circumstancesmentioned above, and if it does not depend on any others, we may write

f(R, D, S, a, e, p, e, e, r', r" ) =o. (37)

We have, in this instance, m = 7, k = 4 hence i = 3 and the equationwill be reducible to the form

y(11„11„11„tt, r', r", . ) = o. (38)

It shortens the process of solution to remark that S'pe and Da/S' are

both dimensionless, so that as there can be at most only three such

independent dimensionless products, only one remains to be found and

these two can be used for two of the II's, whatever form of solution we

choose.Since we wish ultimately to solve for R we must make [R] = [P] in

finding the third II, and the four [Q]'s are to be selected from among

[D, S, a, e, e, p]. Since the two relations [Sepe]=[i] and [DaS ']=[t]are already given among the 6 units, these two combinations can not

occur and there are only 6 combinations of 4 independent units each,

instead of the z5 which there would otherwise be. Using these 6 com-

binations, together with [R] = [P], successively in an equation of the

form (t4), we find that the resulting forms obtained for our third II are

e'S'p,II=~ e'S e'alp e'a&

RD'» ' RD~ ' RDI» 'e'a'p, e'a'

RS ' RS'» (39)

The 6 resulting forms of equation (38) are all equivalent, and it is sufficient

to consider any one of them, e. g. ,

t esca's~ DaS'pe, —,tt, ,r', r",

[= o,

l RS(4o)

Vor..IV.No. g. ON PHYSICALLY SIMILAR SYSTEMS. 363

which we may put into the form

8~coy ' Dcs ( S se, ~ e, r', r", (4t)

Since D, a, and S appear in the unknown function, this equation givesus no definite information except that R is proportional to e', and we cannot tell anything about how R may depend on a or S. It is thereforeevident that we must assume a more limited range of dependence of Rif we are to get any more definite results. If we assume that R doesnot depend on the linear size of the charge, p must be independent of D,therefore of Da/S', and therefore of a. Hence on this hypothesis equation

(4t) reduces togmge

R = sg (S'y e8, r', r", . ). (4z)

If we assume that R is independent of S but make no assumption as toD; we can not eliminate a from the unknown function because S appears,in equation (4t), in two of the independent arguments of y. If we assume

that R does not depend on either S or D, y& of equation (42) must containS as a factor and therefore Sp,~a~, so that the equation reduces to

R = e'as'&e&rpr(8, r', r", ~ ). (43)

Either (4z) or (4g) might have been obtained by making the necessaryexclusion of variables from equation (37) and working the result outseparately. It is, however, much more instructive to include, at thestart, all the quantities which we can reasonably suppose might be ofimportance, and then carry out our exclusion after the general equation

(4r) has been obtained.If we now restrict our considerations to a charge of fixed amount and

of fixed shape and size; and if we further suppose it to move always in

the same medium, e, D, p, , ~, r', r", etc. , are all constant and equations

(4r), (4z), and (43) degenerate into

R = a'f&(S, a, 8),R = a'fs(S, e),R = a'fs(8)

(4t, a)(4z, a)(43, a)

These are the equations for the radiation of an electron of specified

shape and size moving in free space. As we see, the simple form (43, a)can not be obtained without assumptions which are far from plausible.

9. Thernuzl Transmissivity. —For variety, we may illustrate the use

of the same general method by applying it to a thermal problem, namely

that of the transmission of heat between the wall of a metal pipe and a

stream of Huid which is flowing through it and is hotter or colder thanthe pipe. Although a great many experiments have been made on thisimportant practical subject, our information is still very incomplete,and the method of dimensions may be of service, both in planning ex-

periments, and in analyzing and interpreting the results obtained.Let the pipe be of uniform section and iong compared with its greatest

diameter. Let the shape of its section be specihed by a number of length

ratios, which we will represent by a single symbol r, and let D be any one

dimension, such as the diameter if the pipe is round. Let 5 be the mean

linear speed of the fluid at any section, as measured by the rate of dis-

charge. Let 8 be the absolute temperature of the wa11 surface at anysection, and 68 the difference between this and the mean temperature

of the Huid at that section. There will be a How of heat between the

pipe and the Huid in one direction or the other, according to the direction

of the temperature drop, and until the contrary is shown we must assume

that this rate of heat transmission may depend on D, 5, 8, 58, and the

properties of the Huid.

We shall suppose that the part played by radiation is negligible, thereby

excluding the consideration of such cases as fame in boiler tubes; and

the thermal properties of the fluid which need attention are then its

thermal conductivity X'and its specihc heat C. The rate of transmission

will, in general, be affected by convection, so that we must take account

of the mechanical properties which determine the nature of the motion

of the Huid, namely its density p and viscosity p. If the fluid is a gas,

the compressibility may also need to be taken into consideration; but

it appears that at speeds which are less than one.half that of sound in

the medium, this element may be disregarded, gases behaving sensibly

like liquids of the same density and viscosity. We shall limit our

considerations to these moderate speeds, so that such results as are

obtained will not be applicable without modification to the transmission

of heat between a steam-turbine nozzle and the jet flowing through it,

or to similar cases where the speed is very high.

Let vh8 be the heat transmitted per unit time through unit area of wall

surface, v being known as the transmission coeScient or " transmissivity. "To obviate the need of introducing the mechanical equivalent of heat,

%e may suppose quantities of heat to be measured in absolute work units

derived from the fundamental mechanical units [et, i, r] wluch, together

with the temperature unit [8],wi11 su%ce for all the quantities with which

we have to deal. %'e do net assume that the transmissivity r is inde-

pendent of 58; that question is left open.

If we have not overlooked any of the circumstances which have a

Vox, . IV.No. 4. ON PHFSICALLF SIMILAR SFSTZMS.

sensible effect on the heat transmission, we may now write

f(r, D, S, '8, p, p, X, C, 58/8, r) =o (~)8 and 68 being quantities of the same kind, so that only one of themappears in the list of variables while the other is represented by a ratio,'8/8. The number of diferent kinds of quantity is n = 8; the number

of fundamental units required is k = 4; hence i = 4 and the equation,whatever its precise form, must be reducible to

y(n„n„a„n„'8/8,r) = o, (45)

We wish to hnd an expression for the transmissivity; hence in hnding

the II's by means of equation (i4), r must be one of the P's. In theprocess of solution 1et us set

[p, D, S, 68] = [Q~, Qq, Q3, Q4],

[r, p, X, C] =[Pg, P2, Pi, P,]

The dimensions of these quantities on the [m, I, I, 8] system are

[p] = [mf-'],

[D] = [I],

Pl=[»'],[~8] = [8],

[r] = [81I '8 ']

[p] = [ml-'' ']

[~] = [nr» '8-'], -[C] = [PI '8'],

and if we use these values in solving equations (r4) for the four II's, the

usual routine procedure gives us the equation

&vh8 p, X68IpS'' pDS' pDH'

CA8 68S '

8' r =o (46)

pP ~ p, M8 C68, 6868 'IpDS'pDS'' S 8' (4~)

as a form to which the equation for v must be reducible if our initial

assumptions regarding the dependence of v on the other quantities were

correct.Equation (47) conveys no definite information whatever, but we may

give a few indications of how such an equation may, nevertheless, be

utilized in supplementiag incomplete experimental data or in planning

new experiments.Since dimensional reasoning can give us no further help, we turn to

experiment. It is known that while at low speeds we may have stream-

line motion of a Quid through a smooth straight pipe, this form of motion

becomes unstable at higher sp .eds and breaks up into turbulent 6 otion.AVe shall suppose that the spa d is high enough and the pipe suf' .ientlyrough that the motion of the H .id is very turbulent. It is known, further,that under these conditions the mechanical behavior of the Huis and thenature of its motion are nearly independent of the value of the viscosity.And since in turbulent motion, convection will certainly play an important

part in the phenomenon of heat transmission, and the nature of the Huid

motion will therefore be important, it is legitimate to assume, as an

approximation at all events, that p, does not appear in the equation for v.

which applies to these conditions of flow. The variable p/pDS will

therefore be absent and equation (47) will assume the simpler form

p5' / XA8 C68 58~e "' PDS3 S e

We must now resort to experiments on transmission, for information

about the form of y, varying the arguments of p separately and deter-

mining corresponding values of v. To vary one of these arguments we

have to vary its separate factors, which are the physical quantities over

which we have direct control; and it is usually most convenient in practiceto vary these separate factors one at a time, —for instance, to 6nd the

relation of T to S when everything else is kept constant. If we are tovary a single one of the arguments of q by varying a particular one of

the physical quantities in question, that quantity must appear in only

one of the arguments. This is true, in equation (48), of p, X, C, and 8,

so that we could proceed at once to investigate the form of y by making

experiments on the relations of 7- to these quantities separately. On

the other hand, D, S, and 60 appear in more than one argument, so that

we could not at once interpret the results of experiments in which one of

these quantities was varied.Now it is not practicable to vary the density, conductivity, and specihc

heat of the Huid arbitrarily and independently, though we may keep

them all constant by making all our experiments on the same Huid.

Furthermore, while 8 may be varied independently of D, 5, and 68, 8

inevitably inHuences the properties of the Huid, which can not be kept

entirely constant during variations of temperature; and, in addition,

attempts to vary 8 over a wide range may encounter formidable difhcul-

ties. The quantities p, ), C, and 8 are thus precisely the ones which we

do not want to use as independent variables. In practice, the most

natural and convenient mode of experimentation is to.vary 5 or 68; and

if we have various pipes available, D also may be varied. Hence equation

(48) is not at present in a suitable form for our purpose and the argu-

VOL. Iv.Xo. 4. O.'7 PH YSICALLY SI.l1ILA R SYSTI:lIS. 367

ments or y must be replaced by others wh'. ch are still independent anddimensi bless but in xvhich D, S, and 58 art, if possible, separated.

Before proceeding to this transformation. ., ave shall first limit our con-siderations to pipes of a particular shape, e. g. , round pipes, which «re thethe most important. The ratios r are then constant, and so long «s it isunderstood that we refer only to round pipes, r may be omitted froni theequations, the effect of varying shape being left for separate investing«-tion, after the study has been completed for round pipes.

Since the speed S is the easiest of our qu«ntities to vary «rbitr«rily,we attend to it first: it appears in two of the «rguments of p «nd we will

therefore replace one of these by another which does not cont«in S.Since the form of y is unknown, we may raise any one of its «rgumcntsto any desired power. AVe take the 2, '3 power of the fir~ «nd notice that

Let

K being then a quantity which involves only properties of the fluid and

may be regarded as one of its characteristic constants. 9,'e noway now

write

But any function of xy and y may be expressed as a function of x and

y or of x" and y": hence we may replace (g8) by the equivalent equation,referring to a fixed shape of cross section,

in which D, S and 8 are separated, though b, tII remains involved in all three

arguments. Equation (49) is suitable for the interpretation of experi-

mental data on 7- obtained by varying D, S, and 8 separately, for thevariations will vary the three arguments of p& separately and so tell us

how y& varies with the whole of any one of its three arguments.

A sufficiently complete and accurate experimental investigation of this

sort would, in principle, always enable us to find the complete form of the

operator y~, and the use of dimensional reasoning has the advantage

that it enables us to plan the experiments rationally. It may turn out

that the form of y~ is so complicated and the investigation so laborious

E. BUCKINGHAM. tSECONDSERIES.

that a complete solution of the problem is virtually impossible, or theresult to be obtained not of sufhcient importance to warrant the laborinvolved. In such instances the use of the principle of similarity, i . e. ,

reducing the unknown function to a constant by keeping all its argumentsconstant, sometimes permits of our securing partial information which

suffices for particular practical purposes, and this will be illustrated in

the following section. On the other hand, especially in cases where thequantities in question can not be measured very accurately or where no

great accuracy in the results is required, it may happen that, to theapproximation needed, the form of the unknown function is very simple.The method of procedure in such an instance may be illustrated by con-

tinuing the consideration of transmissi~ity.Returning to equation (g9), let us consider variations of the speed S.

It appears from experiment that the transmissi~ ity is nearly proportional

to the o.8 power of the speed when other things are constant; and merely

to illustrate how such a result might be used, we shall suppose this

relation to be exact. It follows that 5 can not be involved in yi exceptas a factor S ". And since S appears only in the argument CD8/S',

rp& must contain the factor (Chg/S')". Equation (g9) must therefore

have the more specific form

7 = pSO ~/~ ~/go ~/2(K/D2/g gg/g)

This is simpler than before and suitable for continuing the work by varying

the diameter and wall temperature of the pipe, or by using various Huids

so as to vary E, the va lues of p, X, and C being assumed to be known for

the fluids used.XVe will suppose, however, that it is not practicable to vary 8 through

any wide range and that we prefer to make experiments with various

values of the temperature drop 50 before altering D, which requires the

dismantling of the apparatus and the substitution of a new pipe of dif-

ferent diameter. We must then transform equation (5o) in such a way

that 68, which we are to use experimentally as the independent variable,

appears in only one of the arguments of the unknown function. This

is evidently accomplished by writing

T = pS0.8cl.lgg0. 1~3(K/D2+g K/DRg)

which is suited to the interpretation of experiments on the dependence

of r on 6,8.It is commonly assumed that so long as 68 is small, r is independent of

48. If experiment were to show that this relation was a general one, it

would thereby be proved that pl must contain (K/D'6g)" as a factor,

vo~. Iv.No. 4. 0V PHYSICALLY SIMILAR SYSTEMS. 369

and equation (5s) would receive the still more specific form

pSo'sC '&Eo.r

D" "' iD' i

Having reached this point, the investigation might be completed by

varying D, or changing the fluid so as to vary the value of E, the two

methods providing a mutual check. If 8 also were varied, a second

check would be provided.

If, to take a purely hypothetical case, it were found that, in pipes for

which r is sensibly proportional to S", and within temperature limits

such that r is sensibly independent of 68, r was also sensibly independent

of the diameter D, we should know that within these limits y~ could be

represented with sensible accuracy by (D"-0''k) ' )& constant and r by

the equationr = const. )( pS sC""8".

Or if, to take another imaginary result, it were found that the trans-

missivity, beside being independent of the viscosity, proportional to the

o.8 power of the speed, and independent of the temperature diNerence,

w'as also independent of the temperature of the wall surface, we should

know that the expression for r must be

O.sSo.8Co.sgo.2

r = const. X

It would be out of place here to pursue this subject into an analysis

of the numerous but unhomogeneous data which have been published

concerning transmissivity. Enough has been said to illustrate the pro-

cedure and to show that the utility of the dimensional method is by no

means confined to its applications to hydrodynamics or electromagnetic

theory.to. An Illustration of Dynamical Similarity The ap.p—lication of di-

mensional reasoning to mechanical problems is often useful in the inter-

pretation of model experiments designed to furnish, at a comparatively

small expense, information about the performance to be expected from

full-sized machines. Advantage is then taken of the idea of dynamical

similarity —a particular case of physical similarity in general. Since

this subject seems to be less &amiliar to physicists than it deserves to be,

a single illustrative example may, perhaps, be worth giving.

It was found, in section g, that if the thrust F of a screw propeller of

given shape and immersion can be assumed to depend only on the di-

ameter D, the speed of advance S, the number of turns per unit time n,

the density and viscosity of the liquid p and p, , and the acceleration of

370 E. BUCKINGHAM. Sacowp

gravity g, we must have the relation givell by equation (20) or

,f De pDS DgZ =,DS�&-~ii' ~ 5 (53)

in which the ratios r specify the shape and immersion of the propeller.The principle of dynamical similarity states that in passing from one

screw propeller to a second, in the same or in another liquid, any threekinds of quantity, such as (p, D, S), which can provide fundamental

units, may be changed in any ratios whatever; and that the equationwhich connects the thrust with the other quantities will remain precisely

the same if the values of the arguments of y remain unchanged. Thismeans, in simpler language, that if we hnd the value of the constant .Vin the equation

from an experiment in which the arguments of y have a certain 6xeci set

of values, the same constant is applicable to any values of (p, D, 5) if

the values of Dn/S, pDS/Ii, Dg/S', and the r's are the same in the second

case as in the hrst.The simplest of the requirements for the useful application of equation

(53) is that the r's shall be constant; hence the two propellers, whatever

their diameters, must be geometrically similar and similarly immersed;

and the smaller may be called the model while the larger is called the

original. The next simplest condition is that Dn 5 shall remain constant.

Now «Dn is the speed of the circumferential motion of a point on the

tip of one of the blades, and ~(Dn//S) is the tangent of the angle between

the actual helical path of such a point and the direction of advance of

the screw as a whole, which is supposed to coincide with the axis of the

server. The blades being of a 6xed shape, the condition that De'5

shall be constant is the same as the condition that the "angle of attack "of the blades on the still water into which they are advancing shall be

constant. If p is the pitch of the propeller so that pe is the so-railed

"speed of the screw "or the speed at which it would advance if the water

acted like a sohd nlrftg (Ps —S) ls tile slip slid (pa S)/pll ls the

"slip ratio. " It is easily seen that if Ds jS is constant for propellers of

a given shape, the slip ratio is constant. Our two conditions may now be

expressed by saying that for two server propellers to be dynamically

sixnilar, they must 6rst of all have the same shape and be run at the same

relative ixnmersion and at the same slip ratio.

When the foregoing preliminary conditions are fulfilled, equation (g3)

reduces to the form

(se)

and the next question is whether we can obtain any information aboutthe thrust to be expected from a screw of diameter D run at the speed S,by experiments on a model screw of diameter D', run at the speed S'and at the same immersion and slip ratio as the original. The answer

depends on our ability to arrange matters so that pDS, 'p and Dg/Sshall be the same in the model experiment as in the practical operationof the fuH-sized original, and we at once encounter di%culties. In the

first place, the intensity of' gravity g is sensibly constant so that D/5must also be kept constant. But on the other hand, we are virtually

limited to experimenting in water for which p/y is sensibly constant.

Hence DS as well as D/5' must be kept constant, so that neither D nor

5 can be varied: in other words, we can not, in practice, run a mduced-

scale model screw propeller so that it shall be dynamically similar to its

original. Qfe must therefore limit ourselves to a less ambitious program

and attempt to obtain an approximate result which may be of some

value, even though it is recognized as incomplete; and to do this we must

find a plausible pretext for omitting one of the two arguments of y from

equation (gg).This presents no difhculty; For it is apparent from various hydro-

dynamic experiments tha& when a quid is in very turbulent motion its

mechanical behaa ior is litt1e inHuenec~l by viscosity, density being much

more important. How the motion of the water about the blades of a

scrcm propeller at ordinary working speeds is certainly very turbulent

indra, so that we may safely assume that if pr, , i. e., @AS pa, occurs at all

in equation (~), it is only in terms with very small exponents. It is

therefore a legitimate approximation to omit it altogether and write the

equation in the simpler form

Since gravity is sensibly constant, we can now make two propeBers

dynamically similar, if they satisfy the preliminary conditions regarding

shape, immersion, and slip ratio, by running them at speeds such that

D P is constant. The condition for "corresponding speeds" is there»

fore

accented letters refernng to the model and unaccented to the original.

5'hen the two are run at corresponding speeds we therefore have, by

equation ($$),P ~ ra»P P' &D'f

D7 E. BUCKID GHA M. tSECONDSERIES.

If the model is run in water of the same density as that in which the full-

sized propeller is to run, p = p' and we have

, &D&'F = F'I —

I.

i D'i

If a propeller is very deeply immersed so that no disturbance of thewater surface is produced, the weight of the water can have no influence

on the thrust and g can not appear in equation (55). The unknown

function then degenerates into a mere constant and the equation reduces

toF = XpD'S'

Any two propellers are then dynamically similar, whatever their speeds,if they have the same shape and are run at the same slip ratio, so that we

have, for very deep immersion in a given liquid,

F

WADS

By disregarding viscosity we have, in effect, disregarded the effect of

skin. friction on the action of the propeller; and we have also left aside

the question of cavitation. But without venturing further into the

chaos of screw-propeller theory, the foregoing example will serve to

illustrate the sort of use that may be made of dimensional reasoning in

attacking mechanical problems which ar- like most of those that occur

in prac+. ical hydro- and aerodynamics —too dificult to be handled at all

by ordinary methods.r t. The Relation of lite Laro of Gravitation to Oar Ordinary System of

mechanical Units. —In our reasoning up to the present point, it has been

assumed that three fundamental, i. e. , independent, units are required

in an absolute system for measuring all the kinds of quantity needed in

the description of purely mechanical phenomena, two more being required

for thermal and electromagnetic quantities. If this assumption is

permissible, a purely mechanical system may be kept similar to itself

when any three independent kinds of mechanical quantity pertaining

to it are varied in arbitrary ratios, by simultaneously changing the re-

maining kinds of quantity in ratios specified by equation (t4), as de-

scribed in section 6. %e must now examine this assumption.

9,'hen we say that one quantity is derived from another or others which

act as fundamental, we mean that by using or combining particular

examples of these other kinds of quantitity in some specified manner, we

can fix a quantity of the derived kind which has a particular definite

Vo~. !V.No. 4. OX PHYSICALLY Sl.lflL.4R SE STE.lfS.

magnitude. For instance, we derive a unit ot force from independentfundament il units of mass, length, and time, by usiiig these iinits in acertain way which is fixed by definition, and ive thereby det«rniin«adefinite force which is reproducible and nial be used as;i uiiit. X&iw

by Xewton's laiv of gravitation it is, in principle, possilile to derive oneof the three fundamental units of mech;inics froni the other two. I.ettwo free masses be placed at rest at a distance apart which is

uteri

1;irge

compared with their linear dimensions. Lct t1iem be released «nd 'illow«cl

to approach each other by a certain mcasur«d dist ince, and let the tinge

required to cover this distance be obs«rved. Tliis interv;il of tinie is

fixed by the masses and the distances: in otlicr ivords, an int«rv;il oftime can be derived from masses and lengtlis, ind by «dopting a suit. ;ilile

form of definition, a unit of time can be d«rived fro»i the unit. s of »i;is»

and length. It is, of course, imniaterial wliicli one of tlie tlir««units is

derived from the other two; the point is tli;it if we utilize the I;i' of

gravitation, only two fund;imental units ar«ne«d«d for ni««1i;i»ical

quantities, instead of the three ivhich phd. sicists ordin;irili iis«. Bycarrying out this process or some other cqiiiv;ilent to it, w«s}ii~iilil «li-

minate one of our three primary standards, —the iiitcrii;ition;il kilfiyrani,

the international meter, or the standard clock, na»icly t)ie rotati»g «arth

which preserves the mean solar second. & or practic;il piirpose» ivc

should still use these three standards, but one of then& xvoulcl be redu««if

to the rank of a secondary or working standard.

One reason for not proceeding in this manner is th;it we do iiot yetknow the value of the gravitation constant accurately enough to bring

the proposition within the range of practicability. But siiice we iiiust

admit the theoretical possibility of such a procedure if we reciignize tlie

law of gravitation, it is incumbent upon us to consider what bearirig

this possibility may have on our dimensional reasoning and on our appli-

cations of the theorem of physical similarity; for the numb«r of funda-

mental units needed is a matter of vital importance to our conclusions

regarding any practical problem. For example; in treating t)ie screw

propeller, we assumed that [ng, l, t] were independent units and therefore

that two propellers could be made to constitute dynamically si»filar

systems when three quantities p, D, and S were varied in «rbitrary r;itios

upon passing from one system to the other. The question now evidently

presents itself: ought we not io have limited the arbitrary variations totwo; are we not bound to treat mechanical quantities as derived from only

two and not from three independent fundamental quantities. 'To see the answer to this question, we may read over again the definition

of physically similar systems given in section 6. It was found that a

374 E. BUCKINGHAM. tSECOWnSERIES.

physical system remains similar to itself, as regards any relation amonga number of kinds of quantity, when certain of these kinds —equal innumber to the fundamental units required for the absolute measurementof all the quantities involved in the relation —are subject to variation inarbitrary ratios, if we fix the ratios in which the remaining kinds of quan-tity shall then change by imposing the condition that the II s shallremain invariable. 9'e now see that the answer to the question: howmany fundamental mechanical units are to be used? i. e. , to the questionwhether we are or are not at liberty to ignore the law of gravitation,depends on the nature of the relation in question. If the relation withwhich we happen to be concerned refers to and characterizes some phe-nomenon which does not involve and is not affected by the form of thelaw of gravitation, we can carry out a complete investigation of thephenomenon and represent our results by a complete equation withoutever knowing of the existence of the law of gravitation: this law does notconcern us, and our knowledge of the phenomenon under investigationdoes not depend on our knowing the correct expression for. the law ofgravitation. AVe are therefore plainly at liberty to ignore it altogether,and if we do so, three fundamental units are indispensable because the

only means of eliminating one of them is to use the law of gravitation.It is not necessary that the phenomenon be unaffected by the weight ofmaterial bodies, but merely that it be not sensibly dependent on thefact that weight is proportional to the mass of the Earth and to the inverse

square of the distance from its center.In the most general case, when we include within the field of our reason-

ing all kinds of physical quantity and all Possible relations among them,

we must admit our familiarity with the law of gravitation and limit

ourselves to two fundamental mechanical units. But if for " all possible

relations" we substitute "all relations that do not involve the law of

gravitation, "we may ignore the law and proceed as if it were non-existent.

With this single proviso all our foregoing reasoning retains its full

validity. The limitation is seldom felt, because, in practice, physicists

are seldom concerned with the law of gravitation: for all our ordinary

physical phenomena occur subject to the attraction of an earth of constant

mass and most of them occur under such circumstances that the variation

of gravity with height is of no sensible importance. In precise geodesy

and still more in astronomy, the observed phenomena do involve the

operation of the law of gravitation in such a way that they can not be

completely described without making explicit use of it. If the physical

relations which characterize such phenomena are under discussion, we

must recognize the law of gravitation, we must regard all mechanical

vol. . Iv.No. 4. ON PHYSICALLY SIMILAR SYSTEMS.

units as derivable from two and not three independent fundamentalunits, and if a physical system is to remain similar to itself only fourand not five arbitrary changes are possible, or if we exclude thermal andelectromagnetic quantities, only two. The geodesist and the astronomermust therefore, in using dimensional reasoning, submit to one restrictionfrom which the physicist is usually free, though this formal restriction isoffset by the power of using the law of gravitation explicitly.

To take an illustration, let us suppose that we have to consider aphenomenon which involves mechanical and electromagnetic but notthermal quantities, and that the law of gravitation in its general formdoes not influence the phenomenon. The physical system in which thisphenomenon occurs may remain similar to itself while four independentkinds of quantity Q are changed in any four arbitrary ratios, if all theother kinds P involved in the phenomenon are changed in the ratiosspecified by equation (i4) taken with the arbitrary changes of the Q's.

Ke may, for example, divide all lengths by x, divide all times by x,multiply all masses by x, and leave all electrical charges unchanged: thealtered system will be similar to the original one as regards all phenomenathat do not depend on the law of gravitation, if the remaining kinds ofquantity are changed as shown by equation (I4). But if the phenomenoninvolves the law of gravitation we can impose only three arbitrary ratiosof change, of which one must refer to purely electromagnetic quantities:we can no longer impose arbitrary conditions on lengths, times, and massesbut only on two of these kinds of magnitude. To put it in another way,and omitting electromagnetic quantities, which so far as we know have

nothing to do with the case in hand, we may keep a gravitational systemsimilar to itself while we change its size and its time intervals in anyarbitrary ratios; but after the change, corresponding gravitational forcesmust stand in a determinate ratio which is not arbitrary. Or to makeit less abstract, if we construct a miniature universe by multiplying all

actual lengths by a, and if we change the densities in such a way that themass of every volume element of the miniature universe is b times themass of the corresponding volume element of the actual universe, then

if the miniature universe is to be mechanically similar to the actualuniverse, the gravitational forces in the miniature universe must bearto the corresponding gravitational forces in the actual universe a ratiofixed by the law of gravitat on. And if the speeds at which gravitationalphenomena occur in the mini'ature universe are to have the same numer-

ical values as corresponding speeds in the actual universe, the unit ortime or speed can not be fixed arbitrarily but must have a particulafrelation to our actual unit.

376 F.. BUCK I%GAIA M. tSscoNnsaaras.

Conclusion. —A convenient summary of the general consequences ofthe principle of dimensional homogeneity consists in the statement thatany equation which describes completely a relation subsisting among anumber of physical quantities of an equal or sm;ilier nunibcr of diferentkinds, is reducible to the form

P(IIr, II., - ~, etc.) = o,

in which the II's are all the independent dimcnsionlcss proc)units of t)ic

form Q&*, Q.", , etc. that can be made by using the syml&ols &&i ail &he

quantities Q.M, hile this thcorcrri appears rather nonconimittal, it is in fact;i poN. ~ rt trl

tool and comparable, in tliis rcganl, to thc r»ct)io&ls of thermo&)~ naming. s

or Lagrangc's nzcthod of gcricr;rlizcrl i.oor&)iriatc~. It is hoped t)i;it t)ic

few sample illustrations of its usc w)ii~. )i )iavc )iicri 3;ii.cri xvi)l proic

interesting to physicists who have riot. )iccn iri t)ic )i;i)iit ot r», ikirig r»trt. .)i

use of dimensional rcaso»ing; )iut. if this paper merel& hei) is a ) it t lc toN. ;rrd

dispelling the metaphysic;il fog t)iat seer»s to bc engulfing us, it ii ill )i;ii c

attained its object.BUREAU OF STANDARDS.

June r8, zgr4.